| L(s) = 1 | − 1.34i·3-s − 5-s + (−1.28 + 2.31i)7-s + 1.20·9-s − 2.44·11-s + 1.57·13-s + 1.34i·15-s − 1.11i·17-s − 8.44i·19-s + (3.10 + 1.71i)21-s − 2.62i·23-s + 25-s − 5.63i·27-s − 3.43i·29-s − 9.70·31-s + ⋯ |
| L(s) = 1 | − 0.774i·3-s − 0.447·5-s + (−0.483 + 0.875i)7-s + 0.400·9-s − 0.738·11-s + 0.435·13-s + 0.346i·15-s − 0.271i·17-s − 1.93i·19-s + (0.677 + 0.374i)21-s − 0.548i·23-s + 0.200·25-s − 1.08i·27-s − 0.637i·29-s − 1.74·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9092650245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9092650245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (1.28 - 2.31i)T \) |
| good | 3 | \( 1 + 1.34iT - 3T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 + 1.11iT - 17T^{2} \) |
| 19 | \( 1 + 8.44iT - 19T^{2} \) |
| 23 | \( 1 + 2.62iT - 23T^{2} \) |
| 29 | \( 1 + 3.43iT - 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 6.22iT - 37T^{2} \) |
| 41 | \( 1 + 3.13iT - 41T^{2} \) |
| 43 | \( 1 + 7.45T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 - 6.16iT - 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 + 7.87iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 7.70iT - 79T^{2} \) |
| 83 | \( 1 - 0.813iT - 83T^{2} \) |
| 89 | \( 1 + 5.12iT - 89T^{2} \) |
| 97 | \( 1 - 0.833iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.254244946794353549599352814174, −8.818084143952745713693840775489, −7.68483192195740578383656001409, −7.14938073501085598961720907636, −6.26780526527594257564365358834, −5.34248601622041127095571843006, −4.29500816910004474768993901108, −2.98598865632483876025002454313, −2.10192112610336348079905815373, −0.40182354346054904262165218835,
1.53130466862444561871069703712, 3.45832263365446219713562664134, 3.76926842363277390182628804855, 4.85165814623835796701005650600, 5.80027203781318709659229034946, 6.92134766970465248470555368130, 7.67165754340556628752995632588, 8.453460243300605782467651505145, 9.563250070286776202859523450798, 10.18985355463363768102409495009
